alternating series 1

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Question statement

Decide whether the following series is convergent or not

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$\displaystyle\sum_{n=2}^\infty \left( \frac{1}{\sqrt[\var{m}]{n}-1} - \frac{1}{\sqrt[\var{m}]{n}+1}\right)$

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converges

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diverges

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Advice

If you view the series as

$\frac{1}{\sqrt[\var{m}]{2}-1} - \frac{1}{\sqrt[\var{m}]{2}+1} + \frac{1}{\sqrt[\var{m}]{3}-1} - \frac{1}{\sqrt[\var{m}]{3}+1} + \frac{1}{\sqrt[\var{m}]{4}-1} - \frac{1}{\sqrt[\var{m}]{4}+1} + \cdots$

It is an alternating series. However, alternating series does not work (why not?)

Instead, observe that

$\frac{1}{\sqrt[\var{m}]{n}-1} - \frac{1}{\sqrt[\var{m}]{n}+1} = \frac{1}{n^{2/{\var{m}}}-1}.$

Also, observe that

$\frac{1}{n^{2/{\var{m}}}} \leq \frac{1}{n^{2/{\var{m}}}-1}.$

By $p$ -series test, $\displaystyle \sum_{n=2}^\infty \frac{1}{n^{2/{\var{m}}}}$ is divergent (as $2/{\var{m}} \leq 1$ ). Then by comparison test, $\displaystyle \sum_{n=2}^\infty \frac{1}{n^{2/{\var{m}}}-1} = \sum_{n=2}^\infty \left(\frac{1}{\sqrt[\var{m}]{n}-1} - \frac{1}{\sqrt[\var{m}]{n}+1}\right)$ is also divergent.

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This question is used in the following exam:

Series Homework 1 by Ugur Efem in Ugur's workspace.

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Ugur Efem 2 years, 6 months ago

Ugur Efem 2 years, 6 months ago

Ugur Efem 2 years, 6 months ago

Error in variable testing condition

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alternating series 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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History

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Checkpoint description

Ugur Efem 2 years, 6 months ago

Ugur Efem 2 years, 6 months ago

Ugur Efem 2 years, 6 months ago

Error in variable testing condition

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Generated value: number

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Generated value: `number`